3.1551 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=186 \[ -\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4 (a+b x) (d+e x)^2}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^4 (a+b x)}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)} \]

[Out]

(b^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(a + b*x)) + ((b*d - a*e)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(2*e^4*(a + b*x)*(d + e*x)^2) - (3*b*(b*d - a*e)^2*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x)*(d + e*x)) - (3*b^2*(b*d - a*e)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^4*(a + b*x))

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Rubi [A]  time = 0.241246, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^4 (a+b x) (d+e x)^2}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^4 (a+b x)}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^3,x]

[Out]

(b^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(a + b*x)) + ((b*d - a*e)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(2*e^4*(a + b*x)*(d + e*x)^2) - (3*b*(b*d - a*e)^2*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(e^4*(a + b*x)*(d + e*x)) - (3*b^2*(b*d - a*e)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^4*(a + b*x))

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Rubi in Sympy [A]  time = 21.4991, size = 146, normalized size = 0.78 \[ \frac{3 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{3}} + \frac{3 b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{4} \left (a + b x\right )} - \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 e^{2} \left (d + e x\right )} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**3,x)

[Out]

3*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/e**3 + 3*b**2*(a*e - b*d)*sqrt(a**2 + 2*
a*b*x + b**2*x**2)*log(d + e*x)/(e**4*(a + b*x)) - b*(3*a + 3*b*x)*sqrt(a**2 + 2
*a*b*x + b**2*x**2)/(2*e**2*(d + e*x)) - (a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(2*
e*(d + e*x)**2)

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Mathematica [A]  time = 0.135197, size = 131, normalized size = 0.7 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3 e^3+3 a^2 b e^2 (d+2 e x)-3 a b^2 d e (3 d+4 e x)+6 b^2 (d+e x)^2 (b d-a e) \log (d+e x)+b^3 \left (5 d^3+4 d^2 e x-4 d e^2 x^2-2 e^3 x^3\right )\right )}{2 e^4 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^3,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^3*e^3 + 3*a^2*b*e^2*(d + 2*e*x) - 3*a*b^2*d*e*(3*d + 4*e*
x) + b^3*(5*d^3 + 4*d^2*e*x - 4*d*e^2*x^2 - 2*e^3*x^3) + 6*b^2*(b*d - a*e)*(d +
e*x)^2*Log[d + e*x]))/(2*e^4*(a + b*x)*(d + e*x)^2)

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Maple [A]  time = 0.021, size = 219, normalized size = 1.2 \[{\frac{6\,\ln \left ( ex+d \right ){x}^{2}a{b}^{2}{e}^{3}-6\,\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{2}+2\,{x}^{3}{b}^{3}{e}^{3}+12\,\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{2}-12\,\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}e+4\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}e-6\,\ln \left ( ex+d \right ){b}^{3}{d}^{3}-6\,x{a}^{2}b{e}^{3}+12\,xa{b}^{2}d{e}^{2}-4\,x{b}^{3}{d}^{2}e-{a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+9\,a{b}^{2}{d}^{2}e-5\,{b}^{3}{d}^{3}}{2\, \left ( bx+a \right ) ^{3}{e}^{4} \left ( ex+d \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^3,x)

[Out]

1/2*((b*x+a)^2)^(3/2)*(6*ln(e*x+d)*x^2*a*b^2*e^3-6*ln(e*x+d)*x^2*b^3*d*e^2+2*x^3
*b^3*e^3+12*ln(e*x+d)*x*a*b^2*d*e^2-12*ln(e*x+d)*x*b^3*d^2*e+4*x^2*b^3*d*e^2+6*l
n(e*x+d)*a*b^2*d^2*e-6*ln(e*x+d)*b^3*d^3-6*x*a^2*b*e^3+12*x*a*b^2*d*e^2-4*x*b^3*
d^2*e-a^3*e^3-3*a^2*b*d*e^2+9*a*b^2*d^2*e-5*b^3*d^3)/(b*x+a)^3/e^4/(e*x+d)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.208926, size = 254, normalized size = 1.37 \[ \frac{2 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d e^{2} x^{2} - 5 \, b^{3} d^{3} + 9 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - a^{3} e^{3} - 2 \,{\left (2 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} d^{3} - a b^{2} d^{2} e +{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (b^{3} d^{2} e - a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^3,x, algorithm="fricas")

[Out]

1/2*(2*b^3*e^3*x^3 + 4*b^3*d*e^2*x^2 - 5*b^3*d^3 + 9*a*b^2*d^2*e - 3*a^2*b*d*e^2
- a^3*e^3 - 2*(2*b^3*d^2*e - 6*a*b^2*d*e^2 + 3*a^2*b*e^3)*x - 6*(b^3*d^3 - a*b^
2*d^2*e + (b^3*d*e^2 - a*b^2*e^3)*x^2 + 2*(b^3*d^2*e - a*b^2*d*e^2)*x)*log(e*x +
d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**3,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/(d + e*x)**3, x)

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GIAC/XCAS [A]  time = 0.214025, size = 230, normalized size = 1.24 \[ b^{3} x e^{\left (-3\right )}{\rm sign}\left (b x + a\right ) - 3 \,{\left (b^{3} d{\rm sign}\left (b x + a\right ) - a b^{2} e{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, b^{3} d^{3}{\rm sign}\left (b x + a\right ) - 9 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + a^{3} e^{3}{\rm sign}\left (b x + a\right ) + 6 \,{\left (b^{3} d^{2} e{\rm sign}\left (b x + a\right ) - 2 \, a b^{2} d e^{2}{\rm sign}\left (b x + a\right ) + a^{2} b e^{3}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^3,x, algorithm="giac")

[Out]

b^3*x*e^(-3)*sign(b*x + a) - 3*(b^3*d*sign(b*x + a) - a*b^2*e*sign(b*x + a))*e^(
-4)*ln(abs(x*e + d)) - 1/2*(5*b^3*d^3*sign(b*x + a) - 9*a*b^2*d^2*e*sign(b*x + a
) + 3*a^2*b*d*e^2*sign(b*x + a) + a^3*e^3*sign(b*x + a) + 6*(b^3*d^2*e*sign(b*x
+ a) - 2*a*b^2*d*e^2*sign(b*x + a) + a^2*b*e^3*sign(b*x + a))*x)*e^(-4)/(x*e + d
)^2